Let \(\phi \)  be a \(\Pi _1\) -sentence for \(\in _{\mathbf {NS}_{\omega _1}}\cup \text{{\sf UB}}^V\)  which holds in \(V\) .
Assume that for some forcing notion \(P\) , \(\phi \)  fails in \(V[h]\)  with \(h\)  \(V\) -generic for \(P\) .
By forcing over \(V[h]\)  with the appropriate stationary set preserving (in \(V[h]\) )
forcing notion (using a Woodin cardinal \(\gamma \)  of \(V[h]\) ), we may assume that \(V[h]\)  is extended to a
generic extension \(V[g]\)  such that \(V[g]\)  models \(\mathbf {NS}_{\omega _1}\)  is
saturatedA result of Shelah whose outline can be found in [1]}, or [2]}, or in an handout of Schindler available on his webpage..
Since \(V[g]\)  is an extension of \(V[h]\)  by a stationary set preserving forcing and there are in \(V[h]\)  class many Woodin cardinals, we get that
\(V[h]\sqsubseteq V[g]\)  with respect to the signature \(\in _{\mathbf {NS}_{\omega _1}}\cup \text{{\sf UB}}^V\) .
Since \(\Sigma _1\) -properties are upward absolute and \(\lnot \phi \)  holds in \(V[h]\) ,
\(\phi \)  fails in \(V[g]\)  as well.
Let \(\delta \)  be inaccessible in \(V[g]\)  and let \(\gamma >\delta \)  be a Woodin cardinal.
Let \(G\)  be \(V\) -generic for \(\mathcal {T}^{\omega _1}_\gamma \)  (the countable tower \(\mathbb {Q}_{<\gamma }\)  according to [3]})
and such that \(g\in V[G]\) .
Let \(j_G:V\rightarrow \operatorname{Ult}(V,G)\)  be the induced ultrapower embedding.
Now remark that \(V_\delta [g]\in \operatorname{Ult}(V,G)\)  is \(B^{V[G]}\) -iterable for all
\(B\in \mathsf {UB}^{V}\)  (since
\(V_\eta [g]\in \operatorname{Ult}(V,G)\)  for all \(\eta <\gamma \) , and this suffices to check that \(V_\delta [g]\) 
is \(B^{V[G]}\) -iterable for all \(B\in \mathsf {UB}^V\) , see [4]}).
By [4]} applied in \(\operatorname{Ult}(V,G)\) , there exists in \(\operatorname{Ult}(V,G)\)  an iteration
\(\mathcal {J}=\left\lbrace j_{\alpha \beta }:\alpha \le \beta \le \gamma =\omega _1^{\operatorname{Ult}(V,G)} \right\rbrace \)  of
\(V_\delta [g]\)  such that
\(\mathbf {NS}_{\omega _1}^{X_{\gamma }}=\mathbf {NS}_{\omega _1}^{\operatorname{Ult}(V,G)}\cap X_{\gamma }\) , where
\(X_\alpha =j_{0\alpha }(V_\delta [g])\)  for all \(\alpha \le \gamma =\omega _1^{\operatorname{Ult}(V,G)}\) .
This gives that \(X_{\gamma }\sqsubseteq \operatorname{Ult}(V,G)\)  for \(\in _{\mathbf {NS}_{\omega _1}}\cup \text{{\sf UB}}^V\) .
Since \(V_\delta [g]\models \lnot \phi \) , so does \(X_{\gamma }\) , by elementarity.
But \(\lnot \phi \)  is a \(\Sigma _1\) -sentence, hence it is upward absolute for superstructures, therefore
\(\operatorname{Ult}(V,G)\models \lnot \phi \) . This is a contradiction, since \(\operatorname{Ult}(V,G)\)  is elementarily equivalent to \(V\)  for
\(\in _{\mathbf {NS}_{\omega _1}}\cup \text{{\sf UB}}^V\) , and \(V\models \phi \) .
A similar argument shows that if \(V\)  models a \(\Sigma _1\) -sentence \(\phi \)  for
\(\in _{\mathbf {NS}_{\omega _1}}\cup \text{{\sf UB}}^V\) 
this will remain true in all of its generic extensions:
Assume \(V[h]\models \lnot \phi \) 
for some \(h\)  \(V\) -generic for some forcing notion \(P\in V\) .
Let \(\gamma >|P|\)  be a Woodin cardinal, and let \(g\)  be \(V\) -generic for\(\mathcal {T}_\gamma \)  is the full stationary tower of height \(\gamma \)  whose conditions are
stationary sets in \(V_\gamma \) , denoted as \(\mathbb {P}_{<\gamma }\)  in [3]},
see in particular [3]}.
\(\mathcal {T}_\gamma \)  with \(h\in V[g]\)  and \(\operatorname{crit}(j_g)=\omega _1^V\)  (hence there is in \(g\)  some stationary set of \(V_\gamma \)  concentrating on countable sets).
Then \(V[g]\models \phi \)  since:

\(V_\gamma \models \phi \) , since \(V_\gamma \prec _{1} V\)  for \(\in _{\mathbf {NS}_{\omega _1}}\cup \text{{\sf UB}}^V\) 
by Lemma REF ;

\(V_\gamma ^{\operatorname{Ult}(V,g)}=V_\gamma ^{V[g]}\) , since \(V[g]\)  models that
\(\operatorname{Ult}(V,g)^{<\gamma }\subseteq \operatorname{Ult}(V,g)\) ;

\(V_\gamma ^{\operatorname{Ult}(V,g)}\models \phi \) , by elementarity of \(j_g\) , since
\(j_g(V_\gamma )=V_\gamma ^{\operatorname{Ult}(V,g)}\) ;

\(V_\gamma ^{V[g]}\prec _{\Sigma _1}V[g]\)  with respect to \(\in _{\mathbf {NS}_{\omega _1}}\cup \text{{\sf UB}}^V\) ,
again by Lemma REF  applied in \(V[g]\) .

Now repeat the same argument as before to the \(\Pi _1\) -property \(\lnot \phi \) ,
with \(V[h]\)  in the place of \(V\)  and \(V[g]\)  in the place of \(V[h]\) .
